C.-Y Wang

E A Everett

S C Billups

CiteSeerX

L. T. Watson 
S. C. Billups 
Department of Computer Science, 
Virginia Polytechnic Institute and 
State University, 
Blacksburg, Va. 24061 
C.-Y. Wang 
Departments of Mechanical Engineering 
and Physiology, 
Michigan State University, 
East Lansing, Mich. 48824 
Mem.ASME 
E. A. Everett 
Department of Computer Science, 
University of Wisconsin, 
Madison, Wise. 53706 
Slow Viscous Flow in a Syringe1 
The slow viscous flow in a syringe is modeled by the quasi-steady axisymmetric 
Stokes equation with a point sink for the needle hole. The governing equations are 
approximated using nonstandard finite difference formulas optimized for the 
boundary conditions, and solved numerically using a SOR technique. Streamlines 
and pressure profiles are computed for a variety of syringe configurations. 
1 Introduction 
The syringe is probably the most important and most in-
dispensable instrument in medicine. The syringe consists of 
three parts, the needle, the glass cylinder and the plunger. As 
the plunger is squeezed, the fluid inside the glass cylinder is 
ejected through the hollow needle. 
The present paper is concerned with the low Reynolds 
number flow inside a syringe. This situation occurs in the slow 
controlled intravenous infusion of fluids and drugs in the 
hospital or laboratory. For example, typical values for the 
Reynolds number in indicator-dilution experiments are Re = 
0.07 (50cc syringe at lec/min) or Re = 0.015 (lOcc syringe at 
0.1 cc/min). The infusion rates are kept constant by a 
calibrated infusion pump. 
In Fig. 1 the plunger is moving to the left in a circular 
cylinder. Fluid is forced through a hollow needle of diameter 
usually less than 3 percent of the diameter of the cylinder. Due 
to the geometry the fluid dynamics can be separated into two 
parts: the flow in the needle and the flow in the cylinder. The 
flow in the needle (orifice) has been solved by Dagan, Wein-
baum and Pfeffer [1], who concluded the velocity profile is 
dominantly Poiseuille, with end effects limited to a distance of 
only 1/4 the inner needle diameter. In what follows we shall 
study the cylinder region, with the needle hole approximated 
by a sink. 
1 This work was supported in part by AFOSR Grant 85-0250. 
Contributed by the Bioengineering Division for publication in the JOURNAL OF 
BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Divi-
sion, October 21, 1985; revised manuscript received June 18, 1986. 
2 Formulation 
Let the origin of cylindrical coordinates (r, z) be situated at 
the sink. The cylinder is at z' = 0, r' < a and z' > 0, r ' = a. 
At the time of investigation, the plunger is at z ' = I, r' < a 
traveling with velocity U. Let v' and w' be the velocity com-
ponents in the r' and z ' directions, respectively. 
Since the Reynolds number is much less than one, the flow 
is governed by the quasi-steady axisymmetric Stokes equation 
/ d2 1 9 32 \ 2 
where i/'' is a stream function defined by 
1 d\//' 1 d\j/' 
v w (2) /•' dz' r' dr' 
The boundary conditions for the domain 0 < z ' < /, 0 < r' 
< a are 
o n z ' = 0 , v' =w' = 0, except at the sink at the origin, 
z' =/ , v' =0 , w' = — U 
r' =a, v' = w'=0 
dw 
(3) 
= 0, v dr' 
- = 0 
We normalize all lengths by a, the velocities by U, the 
stream function by - Via2U and drop primes. The governing 
equations become 
/ a2 1 a & \ 2 
VaT2" 7 
 a2 ^ 
" a T * " ^ 
\j/ = Q 
on z = 0, * = 1 , 
~dz~ 
= 0, except at the origin, 
Journal of Biomechanical Engineering NOVEMBER 1986, Vol. 108/317 
Copyright © 1986 by ASME
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Fig. 1 Syringe geometry and coordinate system 
Fig. 2 <Po (Wo = n-1> a n d x lines for /3 = 0.50 
= 0 - : 
a 
/•=!, 
d*P 
= 0, 
1 di/< 
dr \ r 
d^ 
dr 
dip 
dr 
= 2, 
• ) = » , 
(4) 
* = ! , • 
dr 
= 0 
i3 is the instantaneous ratio of the axial length of the 
cylinder to its radius. There exists a weak singularity on the 
circle z = P, r=l where the velocity w is discontinuous and a 
strong singularity at the origin where ip is discontinuous. These 
singularities pose immediate problems in solving for the 
stream function directly. To remove the strong singularity 
analytically, we set 
*P(r,z)=iP0(.r,z)+x(r,z) (5) 
where \pa represents the Stokes flow due to a sink on an in-
finite solid plane [2] 
iP0 = \-z\r
2+z2)'2n (6) 
Thus 
32 1 fl d2 \ 2 
*•) * = 0 (7) ( — -
\dr2 
d 
~d7 + 'dz2 
The boundary conditions are 
dx 
on z = 0, x = 0, 
dz 
= 0 
on z = l X = r
2-l+p3(r2 + (32y3/2, 
°X =3/?2,.2 
dz 
3/32r2(r2+P2)-
(8) 
(9) 
Fig. 3 \p streamlines for /3 
(dashed) 
5.0 (solid) and Poiseuille streamlines 
on /• = = 0, x = 0, Jim - ( J - * L ) = o 
r-o dr \ r dr J 
on r=\, x = zi(\+z2)~V2, 
dx 
dr 
-3z3(l+z2)-
(10) 
(11) 
Although equations (7)-(ll) are regular at the origin, a 
discontinuity in the derivative of x still exists on z = /3, r = 1. 
The boundary conditions are complicated enough such that an 
analytical solution is not worthwhile. In what follows we shall 
use numerical means to solve the problem. 
3 Description of the Numerical Method 
The partial differential equation for x is solved numerically 
using finite difference methods [3], which produce an approx-
imation for x at each grid point. In the middle away from all 
boundaries, standard difference equations are used to approx-
imate the partial derivatives. Near the boundaries, different 
nonstandard formulas are needed. These new difference for-
mulas, using both values and partials of x on the boundaries, 
were derived using the symbolic manipulation package SMP 
[4]. The derivation technique is standard in numerical 
analysis, but because of the high order partials and boundary 
derivative information involved, the mathematical derivation 
is extremely tedious and prone to error. That is why SMP was 
used after an initial manual attempt. To save the reader the ef-
fort involved in reproducing these formulas, all the finite dif-
ference approximations used are given in an appendix. 
The finite difference approximation results in a linear 
system of equations Ax = b, with dimension 2500 for a 
modest 50 x 50 grid. Since the boundary conditions are not 
symmetric and difference formulas using known derivatives 
on the boundaries are employed, the coefficient matrix A is 
not symmetric. The matrix A is nearly diagonally dominant, 
however, suggesting that successive over-relaxation (SOR) 
may work. SOR does not, in fact, converge for all choices of 
the relaxation parameter w anywhere between 0 and 2 (A is not 
symmetric and positive definite). Nevertheless, excellent con-
vergence for SOR was achieved with w = 1.6. 
4 The Streamlines 
After x (r, z) is computed, it is added to \pQ to obtain the 
stream function \p. Figure 2 shows the solution x and \p0 for (3 
= 0.5. Figure 3 shows the sum, the stream function \p, for (3 = 
5 or when the plunger is 5 radius' distance from the sink. For 
such large /3 values, the fluid flow shows 3 distinct 
318/Vol. 108, NOVEMBER 1986 Transactions of the ASME 
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Fig. 4 $ streamlines for 0 = 1.0, (A$ = 0.1) 
1, , 1 
Fig. 5 $ streamlines for /3 = 0.5 and 0 = 0.25 (A^ = 0.1) 
regions: the plunger region at right, the middle Poiseuille 
region and the sink region at left. The flow in the plunger 
region is similar to the entry flow into a circular tube [5], 
where the uniform velocity profile is changed into the 
parabolic profile. Our results agree with Lew and Fung's [5] 
analytical result and Wagner's [6] numerical result for the in-
finite tube at zero Reynolds number. The entrance length is 
about one diameter. The middle regions show almost 
Poiseuille parallel flow. For comparison, the dashed lines are 
the Poiseuille streamlines 
^ = 2 r 2 - r 4 (12) 
In the sink region the fluid turns and flows into the origin. The 
streamlines are dominated by the Stokes sink represented by 
\j/0. There are no regions of recirculation although the velocity 
is quite low in the corner at r = 1, z = 0. If a Moffatt-type ed-
dy [7] does exist, its strength is estimated to be two thousand-
ths of the main flow, and thus undetectable. Figures 4, 5 show 
the changes in streamline patterns as the plunger moves in 
closer. The three distinct regions disappear due to mutual in-
teraction. At (3 = 0.25 the velocity becomes almost radially in-
wards in contrast to the mainly axial Poiseuille flow at /3 = 5. 
5 The Pressure Distribution 
The pressure normalized by Ujx/a is numerically computed 
by finite differences from the normalized Stokes formula: 
Fig. 7 Constant pressure lines for /3 = 1.0 (Ap = 5) 
Fig. 8 Constant pressure lines for /3 = 0.50 (Ap = 30) and 0 = 0.25 (Ap 
= 200) 
dp _ 1 / d V a3i/< \ 1 d V 
~dF~~ 7\dzdr2 + dz3 )+'rr dzdr ' 
(13) 
dp _ i / d3^ ay \ i ay I H 
~Tz T V dr3 + drdz2 ) ~~r> dr2 + r3 dr 
The constant pressure lines are shown in Figs. 6-8. Regions in 
the lower left and upper right corners are not contoured 
because the isobars are too dense. We see a high pressure 
region exists near the corner at z = 0, r = 1 while near the 
center of the plunger (z = /3, r = 0) the pressure shows a local 
high for large /3 and a local low for /3 < 1. In the vicinity of the 
sink, the pressure is high on the surface z = 0 and is low on the 
axis r = 0. The pressure pattern near the sink persists for all 13 
and agrees with the Stokes sink pressure distribution [2]. 
Notice that for (3 = 5 there exists a middle Poiseuille region 
where pressure is independent of radial distance and the 
pressure gradient is uniform axially. 
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6 Discussion 
The eigenfunction expansion method of reference [1] is un-
suitable for the present problem. This is due to the fact that 
the needle hole is so small compared to the cylinder diameter 
that the convergence of the infinite series becomes too slow. If 
the singularity is removed by subtraction as in this paper, then 
eigenf unctions cease to exist. 
The Stokes flow in a syringe is now solved. Although the 
streamlines do not show recirculation, the flow in a syringe is 
not uniform. The pressure distribution is more complicated. 
Regions of local high and local low pressure may cause par-
ticulate matter in the fluid to migrate, resulting in in-
homogeneity of the injectate. 
Stokes flow is not valid in the immediate neighborhood of 
the sink where velocities increase indefinitely. To eliminate the 
singularity, one must consider a hole of finite radius at the 
origin. Dagan, Weinbaum and Pfeffer [1] considered a finite 
cylindrical hole in an infinite plate and computed the 
streamlines and pressure distribution. They found the effect of 
the hole is limited to one hole radius in the vicinity of the junc-
tion. We conclude that the shape and size of the hole has little 
effect on the flow in the syringe presented in this paper. 
What is the force on the plunger? Due to the difference in 
size, almost all hydrodynamic resistance comes from the flow 
through the needle. Reference [1] showed the pressure dif-
ference through a finite sized hole of radius ah and length Lh is 
Finite Difference Formulas 
For completeness, all the finite difference approximations 
used are listed here. These approximations are optimal in the 
sense of using all derivative information available near the 
boundaries. Except for the first few listed, these are all 
nonstandard and had to be derived using SMP [4]. 
For the first partial with respect to r at all points 
dx , , x(z,r+h)-x(z,r-h) 
— (z,r) = 
For the second partial with respect to r at all points 
32x ,- , x(,z,r+h)-2x(z,r)+x(z,r-h) _ _ ( z > f ) = _ 
For the third partial with respect to r in the middle away 
from the boundaries r = 0 and r = 1 
X(z,r + 2h)-2x(z,r + h)+2x(z,r-h)-x(z,r-2h) 
2h3 
Near the boundary r = 0 (i.e., at points {z, h)) 
320/Vol. 108, NOVEMBER 1986 
a linear sum of Poiseuille resistance and Sampson resistance 
[8]. Thus the force on the plunger is a constant 
F = x * 2 ( - i ^ - + 3 ) (14) 
Of course, this formula would not be valid towards the end of 
the injection when the plunger is several needle radii from the 
hole. 
References 
1 Dagan, Z., Weinbaum, S., and Pfeffer, R., "An Infinite-Series Solution 
for the Creeping Motion Through an Orifice of Finite Length," Journal of 
Fluid Mechanics, Vol. 115, 1982, pp. 505-523. 
2 Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, 
Chap. 4, Prentice Hall, N.J., 1965. 
3 Smith, G. D., Numerical Solution of Partial Differential Equations, Ox-
ford University Press, N.Y., 1965. 
4 Wolfram, S., "Symbolic Mathematical Computation," Comm. ACM, 
Vol. 28, 1985, pp. 390-395. 
5 Lew, H. S., and Fung, Y. C , "On the Low-Reynolds Number Entry Flow 
Into a Circular Cylindrical Tube," Journal of Biomechanics, Vol. 2, 1969, pp. 
105-119. 
6 Wagner, M. H., "Developing Flow in Circular Conduits: Transition 
From Plug Flow to Tube Flow," Journal of Fluid Mechanics, Vol. 72, 1975, pp. 
257-268. 
7 Moffatt, H. K., "Viscous and Resistive Eddies Near a Sharp Corner," 
Journal of Fluid Mechanics, Vol. 18, 1964, pp. 1-18. 
8 Sampson, R. A., "On Stoke's Current Function," Philosophical Transac-
tions of the Royal Society of London, Vol. A182, 1891, pp. 449-518. 
33X , , 
- £ 3 - <*•'> = 
~3x(z,r)+2h-^L (Zyr-h)+-^-x(z,r-h)+-^-x(z,r + 2h) 
_ 
Near the boundary r = 1, (i.e., at points (z, 1 - h)) 
d3X . . 
3x(z,r)+2h^- (z,r + h)-~x(z,r + h)-~x(z,r-2h) or 3 3 
• _ 
For the fourth partial with respect to r away from the 
boundaries r = 0 and r = 1 
X(z,r-2h)-4x{z,r-h)+6x(z,r)-4x(z,r + h)+x(z,r+2h) 
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A P P E N D I X 
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Near r = 0, 
-*r {z'r)= 
1 8 113 dx 
16x(z,r) — — x(z,r+3h)+-~ x(z,r + 2h)-9x(z,r + h) ——- x(z,r-h)-5h - ~ - (z,r-h) 4 3 12 dr 
_ 
Near r = 1, 
a4x 
ar4 
(«,r)= 
1 8 113 dx 
16x(z,/-) —— X(z,r~3h) + — x(z,r-2h)-9x(z,r-h) — — x U , r + /i) +5/i —— (z,r + h) 4 3 12 dr 
For the fourth partial with respect to z away from the 
boundaries z = 0 and z = 0 
d*x 
IF-<" ) = 
x(z-2Ar,/-) - 4x (z-£,/•) +6x(z,/") - 4 x ( z + £,r) + x(z + 2fc,r) 
/t4 
Near z = 0, 
d4X 
3z4 
( « , ! • ) = 
16x(z,r) - - J - x(z-3/c,r) + — x ( z - 2 * , r ) - 9 x ( « - * , r ) — ^ x(z + /r,r) + 5 * —— (z + k,r) 4 3 12 dz 
yt4 
Near z = /3 
a4x , , 
dz4 
16x(z,r) — J - x(z-3Ar,r) + — x(«-2* , r ) - 9 X ( z - * , r ) - - ^ x(z + £,r) +5k —*- (z + £,r) 4 3 12 dz 
_ 
For the mixed fourth partial in the middle away from all 
boundaries 
2(30x(z,r)+x(z,r-2h)-\6x{z,r-h)-16x(z,r+h)+x(z,r + 2h)) 
-(30x(z-k,r)+x(z-k,r-2h)-l6x(z-k,r~h)-16x(z-k,r + h)+x(z-k,r+2h)) 
-(30x(z + k,r)+x(z + k,r-2h)-16x(z + k,r-h)-16x(z + k,r + h)+x(z + k,r + 2h)) 
Near z = 0, 
/ - 1 0 257 7 2 
- 2 1 — — x(z,r) +-TTT- x(z-k,r)+-—x(,z + k,r) — — x(z + 2k,r) 
\ 3 144 4 9 
1 5k dX \ 
+ 1 5 - X ( z + 3 * , r ) + — — <*-* , , ) ) 
/ - 1 0 257 7 2 
+ ( — p x ( z , / - - / ! ) + - ^ - x ( z - ^ , / - - / ! ) + — x ( U t / - A ) — — x ( z + 2k,r-h) 
1 5A: ax \ 
Journal of Biomechanical Engineering NOVEMBER 1986, Vol. 108/321 
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/ - 1 0 , 257 7 2 
+ {-Y-x(z,r + h)+-^-x(z-k,r + h)+—x(z + k,r + h)—— x(z + 2k,r + h) 
5k dx 
12 dz 
(z-k,r+h)) + -^-x(z + 3k,r + h) + 
48 
Near z = /? 
„ / - 1 0 257 7 2 
" 2 ( — T ~ * ( z ' r ) + ~ T T r x(z + k,r) + — x(z-k,r) -— x(z-2k,r) 144 
+ _r^" x(z-3A: )r) 
48 12 dz 
U+ £,/•)) 
/ - 1 0 , , 257 7 2 
(—^— X(z,r-h) + - r T r x ( z + * , r - A ) + — x(z-k,r-h) — — x(z-2k,r-h) 144 
48 
x ( z - 3 * , r - A ) - - ^ - - ^ - (« + * , r - A ) ) 
/ - 1 0 , 257 7 2 
+ ( — x ( V + / i ) + - j ^ - x U + t / + / i ) + - - x U - « : / + / i ) — — x ( z - 2 k , r + h) 
1 5/t dx \ 
+ 4 g - x U - 3 * , r + A ) ~ — - J L (z + k,r + h)) 
Near /• = 0, 
- 1 0 „ / -  257 7 2 
-2{—2-x(z,r)+-rn-x(z,r-h)+—x(.z,r+h)——x(z,r + 2h) 144 
5/! dX 1 3/z x \ 
/ - 1 0 257 7 2 
^ - y - x(z-k,r) +-r7r x(z-k,r-h)+—r x(z-k,r + h) — — x(z-k,r + 2h) 144 
1 5h dx \ 
+ —X(z-k,r + 3h)+-^^- (z-k,r-h)) 48 12 dr 
/ - 1 0 257 7 2 
^—3— X(.z + k,r)+-rrrx(z + k,r-h)+—rx(z + k,r + h)—— x(z + k,r + 2h) 144 
1 
(z + k,r-h)j + -X(z + k , r + 3 h ) + ^ - ^ -
Near r = 1, 
„ / - 1 0 257 7 2 
- 2 ( ^ - 3 — X(z,/-) +-TZT X U / + A) + — x(z,r-h) — — x(z,r-2h) 144 
1 
+ l g - x U , r - 3 A ) -
10 
5* dX 
12 dr 
4 
(z,r + A)) 
/ - 1 0 257 7 2 
+ (^—j— X(z-Ar,r) + - T 7 J - X(z-k,r + h) +— x(z-k,r-h) — — x(z-k,r-2h) 144 
+ _ x ( z - * , - 3 A ) - — - £ - < * - * , + *>) 
/ - 1 0 , 257 7 2 
{—j— X(z + k,r) + - ^ ~ x(z + k,r+h) + — x(z + k,r-h) — — x(z + k,r-2h) 
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1 5/7 dX \ 
+i^x(z+k'r-^-^r-ir- (z+k,r+h)) 
48 12 dr / 
For the mixed third partial in the middle away from the 
boundaries z = 0 and z = |8 
30x(z,r~h) +x(z-2k,r-h) - 16x(z-k,r-h) - 16x(z + k,r-h) + x{z + 2k,r-h) 
+ 30x(z,r + h)+x(z-2k,r + h)-l6x(z-k,r+h)-16x(z + k,r+h)+x(z + 2k,r+h) 
Near z = 0, 
^ W <*•'> = 
/ - 1 0 257 7 2 
-\—^-x(.z,r-h)+-^-x(z-k,r-h)+—x(.z+k,r-h)—— x(z + 2k,r~h) 
1 5k dx \ 
+ ̂ x(z + 2k,r-h)+—-~ (z-k,r-h)j 
/ - 1 0 257 , 7 2 
+ ( — p x O v + A) +-J44" x(z-k,r + h)+— x(z + k,r + h) — — x(z + 2k,r + h) 
5k dx 1 k dx \ 
Near z = 0 
d3x 
drdz2 
M 2 ^ ( v ) = 
/ - 10 257 7 2 
- [~Y~ X(z,r-h)+-^- x(z + k,r-h) + — x(z-k,r-h) — — x(z-2k,r~h) 
1 5A: dx \ 
/ - 1 0 257 7 2 
+ (—— x ( v + A ) + - r - — x ( z + k,r + h)+—-x(z-k,r + h)——x(z-2k,r + h) V 3 144 4 9 
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Slow Viscous Flow in a Syringe 1

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Slow Viscous Flow in a Syringe 1

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